Graphs

Example:
We would like to derive from Theorem a formula for the number of
graphs on v vertices. Before we start doing so it should be mentioned
that this example is the second typical example which we present.
The first
examples
were devoted to the
introduction of group theoretical concepts,
to
the derivation of Sylow's Theorem
and to a
number theoretic result of Fermat.
The only example where a suitable choice of GX
and Y was made in order to define and enumerate a mathematical structure
was in fact
the composition of two symmetric groups.
As this example
may have been a little bit artificial we admit that it is
only now that we start to apply our paradigmatic actions more systematically.
Let us see how suitably chosen G, X and Y can be used in order to define
and enumerate the graphs on v vertices.(Later on we shall refine this method
by counting these graphs according to the number of edges or according
to their automorphism group. Finally we shall even use this Ansatz
in order to construct such graphs and also to generate them
uniformly at random.) The way of defining
graphs as orbits of groups may at first glance seem to be circumstantial,
but we shall see that this definition is very flexible since it can easily be
generalized to all other kinds of graphs like multigraphs, directed graphs
and so on.

A labelled (simple) graph
consists
of a set of vertices and
a set of edges
joining pairs of vertices, but neither loops
(i.e. edges joining a vertex with itself) nor multiple edges are allowed.
Thus a labelled graph on v vertices can be considered (after numbering the
vertices from 1 to v, say) as a map f from the set
[v choose 2] of
(unordered) pairs of vertices into the set Y :=2= {0,1 }, where
we put

f( {i,j }):= 1 if an edge joins i and j,

and

f( {i,j }):= 0 otherwise.

For example, the first one of the two labelled graphs of figure
can be identified in this way with the mapping
f :[ 4 choose 2] -> 2 defined by

The symmetric group Sv acts
on v and hence also on
[v choose 2], so that we obtain an action
of Sv on 2 [v choose 2]
which is of the form G(YX) as
was described in the section of paragigmatic examples.
Two labelled graphs are called
isomorphic
if and only if
they lie in the same orbit under this action, i.e. if and only if
each arises
from the other by renumbering the vertices, so that for example the
labelled graphs of figure are isomorphic (apply
p:=(34) ÎS4).

Two isomorphic labelled graphs with 4 vertices

A graphG on v
vertices is defined to be such an isomorphism
class of labelled graphs. It can be visualized by taking any member of the
isomorphism class and deleting the labels.This yields for the
labelled graphs shown in figure the
drawings of figure.

The graphs obtained from the labelled ones above

It should be clear by now what we
mean by a graph, and that in our terminology a graph is
not a pair (V,E) consisting of a set V of vertices and a set E of
edges, but that a graph G can be represented by such a pair,
so that, for example, the graphs of figure
are in fact equal.

If we want to allow multiple edges, say
up to the multiplicity k, then
we again consider X:= [v choose 2], but we change Y into
Y:= {0, ...,k }=k+1.
The elements of

YX=(k+1) [v choose 2]

are called labelled k-graphs
, while by k-graphs
onv vertices
we mean the orbits of Sv on this set.

If we want to allow loops or
multiple loops, we replace X by the
union [v choose 2]Èv, and now f(i)=j,
for i Îv, means that
the vertex with the number i carries a j-fold loop.

If we want to consider digraphs
(i.e. the edges are directed and
neither loops nor parallel edges are allowed), we simply put

X := vi2:= {(i,j) Îv2 | i not = j },

the set of injective pairs over v.

Thus graphs, k-graphs, k-graphs with loops
and also digraphs can be
considered as symmetry classes of mappings. Several other structures
will later on be obtained in the same way. Having defined the graphs
this way we want to count them by an application of
Theorem which
means that we have to derive a formula for c( bar ( p)), the number of
cyclic factors of bar ( p), the permutation induced by pÎSv on the set [v choose 2] of pairs of points,
expressed in terms of the
cycle structure of p. In fact we can do better, we can derive the
cycle type of bar ( p) from the cycle type of p.